

A056944


Amount by which used area of rectangle needed to enclose a nontouching spiral of length n on a square lattice exceeds unused area.


7



0, 1, 2, 2, 2, 4, 3, 2, 4, 6, 4, 2, 4, 6, 8, 5, 2, 4, 6, 8, 10, 6, 2, 4, 6, 8, 10, 12, 7, 2, 4, 6, 8, 10, 12, 14, 8, 2, 4, 6, 8, 10, 12, 14, 16, 9, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 12, 2, 4, 6, 8, 10, 12, 14, 16
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OFFSET

0,3


COMMENTS

m (when n is mth triangular number) followed by m even numbers from 2 through 2m.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000


FORMULA

a(n) = 2n  floor((sqrt(8n+1)1)/2)*ceiling((sqrt(8n+1)1)/2) = 2n  A002024(n)*A003056(n) = 2n  A056942(n) = n A056943(n). If n = t(t+1)/2 then a(n)=t; if n = t(t+1)/2+k with 0 < k <= t then a(n)=2k.


EXAMPLE

a(9)=6 since spiral is as marked by 9 X's in 4*3 = 12 rectangle, with 129 = 3 spaces unused and a usedunused difference of 93 = 6:
X.XX
X..X
XXXX
As a triangle, the first few rows are: 1; 2, 2; 2, 4, 3; 2, 4, 6, 4; 2, 4, 6, 8, 5; 2, 4, 6, 8, 10, 6; 2, 4, 6, 8, 10, 12, 7; ... (= reversal of triangle A143595). Row sums = n^2.  Gary W. Adamson, Aug 26 2008


MATHEMATICA

uar[n_]:=Module[{c=(Sqrt[8n+1]1)/2}, 2nFloor[c]Ceiling[c]]; Array[uar, 90, 0] (* Harvey P. Dale, Aug 14 2013 *)


PROG

(MAGMA) [2*nFloor((Sqrt(8*n+1)1)/2)*Ceiling((Sqrt(8*n+1)1)/2): n in [0..90]]; // Vincenzo Librandi, Aug 06 2017


CROSSREFS

Cf. A002024, A003056, A056942, A056943.
Cf. A143595.  Gary W. Adamson, Aug 26 2008
Sequence in context: A273875 A054709 A121806 * A283681 A222819 A194319
Adjacent sequences: A056941 A056942 A056943 * A056945 A056946 A056947


KEYWORD

easy,nonn,nice


AUTHOR

Henry Bottomley, Jul 13 2000


STATUS

approved



